Letter 246 describes a hypothetical confrontation between a hypothetical Ring-bearing Gandalf and Sauron:
Confrontation of Sauron alone, unaided, self to self was not contemplated. One can imagine
the scene in which Gandalf, say, was placed in such a position. It would be a delicate balance. On one side the true allegiance of the Ring to Sauron; on the other superior strength because Sauron was not actually in possession, and perhaps also because he was weakened by long corruption and expenditure of will in dominating inferiors. If Gandalf proved the victor, the result would have been for Sauron the same as the destruction of the Ring; for him it would have been destroyed, taken from him for ever. But the Ring and all its works would have endured. It would have been the master in the end.
Earlier in the same letter we see just exactly how useful the Ring would have been to a hypothetical Ring-bearing Saruman (or anyone else, for that matter):
Sauron would not have feared the Ring! It was his own and under his will. Even from afar he had an effect upon it, to make it work for its return to himself. In his actual presence none but very few of equal stature could have hoped to withhold it from him.
If Saruman had taken and claimed the Ring for his own, Sauron would have been aware of it immediately. We see this happening when Frodo claims the Ring at the Crack of Doom, and there's no reason to think it would be different for Saruman.
What happens next would depend on Sauron himself; whether he would seek to retake the Ring immediately; whether he would visit Saruman himself or send a lesser servant. In any event, a confrontation between Saruman and Sauron seems as though it would not be a closely fought as the one between Gandalf and Sauron described above; Gandalf was able to cast-out Saruman, so it seems as though Sauron would have more easily had the upper-hand in such a contest.
My own feeling is that since it takes time to learn to master the Ring (also Letter 246), Sauron would have had to pay a visit immediately and before Saruman had the time to achieve this mastery. There would have been no need to send an army; this would have been a one-on-one contest in Orthanc. The question of what Saruman would have done therefore becomes moot since it seems inevitable that Sauron would have been the stronger.
It's interesting here that since both Sauron and Saruman were in origin Maiar of Aule, they may be expected to have some knowledge of each other, and that they would have some idea of each other's strengths and weaknesses. That's information that either could potentially use to tip the balance, but perhaps Saruman would also have had awareness that a contest with Sauron was one that he couldn't possibly win? Or perhaps his lust for power would have blinded him to that? We'll probably never know.